3.172 \(\int e^{2 \tanh ^{-1}(a x)} (c-a c x) \, dx\)

Optimal. Leaf size=13 \[ \frac {1}{2} a c x^2+c x \]

[Out]

c*x+1/2*a*c*x^2

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Rubi [C]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 2.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2288} \[ \frac {c \left (1-a^2 x^2\right ) e^{2 \tanh ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - a*c*x),x]

[Out]

(c*E^(2*ArcTanh[a*x])*(1 - a^2*x^2))/(2*a)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x) \, dx &=\frac {c e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{2 a}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 26, normalized size = 2.00 \[ \frac {c \left (1-a^2 x^2\right ) e^{2 \tanh ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])*(c - a*c*x),x]

[Out]

(c*E^(2*ArcTanh[a*x])*(1 - a^2*x^2))/(2*a)

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fricas [A]  time = 0.50, size = 11, normalized size = 0.85 \[ \frac {1}{2} \, a c x^{2} + c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x, algorithm="fricas")

[Out]

1/2*a*c*x^2 + c*x

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giac [A]  time = 1.27, size = 11, normalized size = 0.85 \[ \frac {1}{2} \, a c x^{2} + c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x, algorithm="giac")

[Out]

1/2*a*c*x^2 + c*x

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maple [A]  time = 0.02, size = 11, normalized size = 0.85 \[ c \left (\frac {1}{2} a \,x^{2}+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x)

[Out]

c*(1/2*a*x^2+x)

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maxima [A]  time = 0.32, size = 11, normalized size = 0.85 \[ \frac {1}{2} \, a c x^{2} + c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a*c*x+c),x, algorithm="maxima")

[Out]

1/2*a*c*x^2 + c*x

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mupad [B]  time = 0.02, size = 9, normalized size = 0.69 \[ \frac {c\,x\,\left (a\,x+2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - a*c*x)*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

(c*x*(a*x + 2))/2

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sympy [A]  time = 0.06, size = 10, normalized size = 0.77 \[ \frac {a c x^{2}}{2} + c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c),x)

[Out]

a*c*x**2/2 + c*x

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